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In previous Fallout games, luck affected many things such as gaining a small boost to all your skills, increasing your crit chance, bigger chance at winning black jack, etc.

In Fallout 4, it seems that there are no crits outside of VATs, there are no longer any skill points, and I couldn't find anywhere to play black jack.

So what exactly does the luck stat do other than to unlock higher level perks?

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Luck does a few things.

First of all, obviously, as noted, it's a prerequisite for several perks. Less obviously, many of these perks have random activations associated with them, such as the XP multiplier from Idiot Savant and the appearance of the Mysterious Stranger. The chance of these events occurring is based on luck.

Secondly, and most importantly for many, is Luck's effect on crits. While they don't occur outside of VATS anymore, in VATS, the rate at which the Critical meter fills up is directly based on luck. More luck, more crits.

Finally, Luck has a small impact on finding things out in the world. Stuff like high value mods, ammo drops, caps in a safe, etc. are all influence by Luck. This effect is small, and probably not worth improving luck for on its own, but it does exist.

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    I also wouldn't be surprised if Luck had a small impact on all of the other random rolls the game does, such as speech checks, but I've not tested this or found it documented yet. – LessPop_MoreFizz Dec 13 '15 at 3:18

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